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Disorder and chaos in
quantum systems II.
Lecture 2.
Boris Altshuler
Physics Department, Columbia University
Previous Lecture:
1. Anderson Localization as Metal-Insulator Transition
Anderson model.
Localized and extended states. Mobility edges.
2. Spectral Statistics and Localization.
Poisson versus Wigner-Dyson.
Anderson transition as a transition between different
types of spectra.
Thouless conductance
P(s)
Conductance g
s
Lecture2.
1. Quantum Chaos,
Integrability and
Localization
P(s)
Particular
nucleus
166Er
P(s)
N. Bohr, Nature
137 (1936) 344.
s
Spectra of
several
nuclei
combined
(after
spacing)
rescaling
by the
mean level
Q:
Why the random matrix
theory (RMT) works so well
for nuclear spectra
Original
answer:
These are systems with a
large number of degrees of
freedom, and therefore
the “complexity” is high
?
there exist very “simple”
Later it
systems with as many as 2
became
degrees of freedom (d=2),
which
demonstrate
RMT
clear that
like spectral statistics
Classical (h =0) Dynamical Systems with d degrees of freedom
Integrable
Systems
The variables can be
separated and the problem
reduces to d onedimensional problems
Examples
1. A ball inside rectangular billiard; d=2
• Vertical motion can be
separated from the
horizontal one
• Vertical and horizontal
components of the
momentum, are both
integrals of motion
2. Circular billiard; d=2
• Radial motion can be
separated from the
angular one
• Angular momentum
and energy are the
integrals of motion
d integrals
of motion
Classical Dynamical Systems with d degrees of freedom
Integrable
The variables can be separated [ d one-dimensional
Systems
problems [d integrals of motion
Rectangular and circular billiard, Kepler problem, . . . ,
1d Hubbard model and other exactly solvable models, . .
Chaotic
Systems
The variables can not be separated [ there is only one
integral of motion - energy
Examples
Sinai billiard
B
Stadium
Kepler problem
in magnetic field
Classical Chaos
h =0
•Nonlinearities
•Exponential dependence on
the original conditions (Lyapunov
exponents)
•Ergodicity
Quantum description of any System
with a finite number of the degrees
of freedom is a linear problem –
Shrodinger equation
Q: What does it mean Quantum Chaos ?
h0
Bohigas – Giannoni – Schmit conjecture
Chaotic
classical analog
Wigner- Dyson
spectral statistics
No quantum
numbers except
energy
Quantum
Classical
Integrable
Chaotic
?
?
Poisson
WignerDyson
Lecture1.
2. Localization
beyond real space
Kolmogorov – Arnold – Moser (KAM) theory
A.N. Kolmogorov,
Dokl. Akad. Nauk
SSSR, 1954.
Proc. 1954 Int.
Congress of
Mathematics, NorthHolland, 1957
Andrey
Kolmogorov
Vladimir
Arnold
Jurgen
Moser
0
Integrable classical Hamiltonian Ĥ 0 , d>1:
Separation of variables: d sets of
action-angle variables
I1 , 1  21t; ... , I 2 , 2  22t;..
1D classical motion – action-angle variables
Kolmogorov – Arnold – Moser (KAM) theory
0
A.N. Kolmogorov,
Dokl. Akad. Nauk
SSSR, 1954.
Proc. 1954 Int.
Congress of
Mathematics, NorthHolland, 1957
Integrable classical Hamiltonian Ĥ 0 , d>1:
Separation of variables: d sets of
action-angle variables
I1 , 1  21t; ... , I 2 , 2  22t;..
Quasiperiodic motion:
set of the frequencies,1 , 2 ,.., d which are
in general incommensurate. Actions I i are
integrals of motion I i t  0
Andrey
Kolmogorov
I1
Vladimir
Arnold
Jurgen
Moser
1

I2
2
…=>
tori
Integrable dynamics:
Each classical trajectory is quasiperiodic
and confined to a particular torus, which
is determined by a set of the integrals of
motion
space
real space
phase space: (x,p)
energy shell
tori
Number of dimensions
d
2d
2d-1
d
For d>1 each torus has measure
zero on the energy shell !
Kolmogorov – Arnold – Moser (KAM) theory
A.N. Kolmogorov, Integrable classical Hamiltonian Ĥ 0 , d>1:
Dokl. Akad. Nauk
SSSR, 1954.
Proc. 1954 Int.
Congress of
Mathematics, NorthHolland, 1957
Separation of variables: d sets of action-angle
variables I ,  2 t;.., I ,  2 t;..
1 1
1
2
2
2
Quasiperiodic motion: set of the frequencies,
1 , 2 ,.., d which are in general incommensurate
Actions
1
I1
Vladimir
Arnold

Q:
Andrey
Kolmogorov
Jurgen
Moser
A:
Ii
are integrals of motion I i
I2
2
t  0
…=>
Will an arbitrary weak perturbation
Vˆ of the integrable Hamiltonian Ĥ 0
destroy the tori and make the motion
ergodic (when each point at the energy
shell will be reached sooner or later)
?
Most of the tori survive KAM
weak and smooth enough theorem
perturbations
Most of the tori survive weak and
smooth enough perturbations
KAM
theorem:
I2
I2
Vˆ  0
Each point in the space of the
integrals of motion corresponds
to a torus and vice versa
I1
Finite motion.
Localization in the space
of the integrals of motion
I1
?
Most of the tori survive weak and
smooth enough perturbations
KAM
theorem:
I2
I2
Vˆ  0
I1
I2
I1
h0
I1
Energy shell
Consider an integrable system.
Each state is characterized by a set of
quantum numbers.
It can be viewed as a point in the space of
quantum numbers. The whole set of the states
forms a lattice in this space.
A perturbation that violates the integrability
provides matrix elements of the hopping
between different sites (Anderson model !?)
Weak enough hopping:
Localization - Poisson
Strong hopping:
transition to Wigner-Dyson
Sinai
billiard
Square
billiard
Localized
momentum space
Disordered
localized
Disordered
extended
extended
Localized
real space
Glossary
Classical
Quantum
Integrable
Integrable


H0  H0 I
Hˆ 0   E   ,

KAM
Localized
Ergodic – distributed all
over the energy shell
Chaotic
Extended ?

  I
Extended Level repulsion, anticrossings,
Wigner-Dyson spectral statistics
states:
Localized Poisson spectral statistics
states:
Invariant
(basis independent)
definition
Chaotic
Integrable
Square
billiard
Disordered
localized
All chaotic
systems
resemble
each other.
All integrable
systems are
integrable in
their own way
Sinai
billiard
Disordered
extended
Consider a finite system of quantum
particles, e.g., fermions. Let the oneparticle spectra be chaotic (WignerDyson).
Q:
What is the statistics of the
many-body spectra?
?
a.The particles do not interact with
each other.
Poisson:
individual energies are conserving quantum
numbers.
b. The particles do interact. ????
Lecture 2.
3. Many-Body
excitation in finite
systems
Decay of a quasiparticle with an energy
Landau Fermi liquid
e
Fermi Sea
e in
Quasiparticle decay rate at
T = 0 in
a clean Fermi Liquid.
I. d=3
e
e
e
Fermi Sea
e
2
e
coupling



 ee e   constant  e F
h
2
d 3
Reasons:
• At small e the energy transfer,  , is small and the integration
over e and  gives the factor e2.
…………………………………………………………………
•The momentum transfer, q , is large and thus the scattering
probability at given e and  does not depend on e ,  or e
Quasiparticle decay rate at
T = 0 in
a clean Fermi Liquid.
II. Low dimensions
Small moments transfer, q , become important at
low dimensions because the scattering probability is
proportional to the squared time of the interaction,
e
1/q
(qvF. )-2
e eF
2
h
 ee e 

e
e
2
e F log e F e 
d 3
d 2
d 1
vF
Quasiparticle
decay
a clean
Quasiparticle
decayrate
rate atatTT= =00inina clean
FermiFermi
Liquid.Liquid.
e
e
e
e
h
 ee e 
e 2 eF
 e 2 e F log e F e 
e
d 3
d 2
d 1
Fermi Sea
Conclusions:
1. For d=3,2 from e  e F
it follows that e ee  h , i.e.,
that the qusiparticles are well determined and the Fermi-liquid
approach is applicable.
2. For d=1
e ee is of the order of h
, i.e., that the Fermi-liquid
approach is not valid for 1d systems of interacting fermions.
Luttinger liquids
Decay of a quasiparticle with an energy e in
Landau Fermi liquid
Quantum dot – zero-dimensional case ?
e
e
e1
e1
Fermi Sea
Decay of a quasiparticle with an energy e in
Landau Fermi liquid
Quantum dot – zero-dimensional case ?
e
e
e1
e1
Fermi Sea
Decay rate of a quasiparticle with energy
( U.Sivan, Y.Imry & A.Aronov,1994 )
Fermi Golden rule:
 e 
  e   1  
 ET 
Mean level
spacing
2
Thouless
energy
e
Decay rate of a quasiparticle with energy
e in 0d.
( U.Sivan, Y.Imry & A.Aronov,1994 )
Fermi Golden rule:
 e 
  e   1  
 ET 
Mean level
spacing
Def:
Zero
dimensional
system
2
Thouless
energy
Recall:
ET
1
g
Thouless
conductance
ET  e  1  g  1
One particle states are
extended all over the system
Decay rate of a quasiparticle with energy
e in 0d.
Problem:
e
zero-dimensional case
e
e1
e1
Fermi Sea
one-particle spectrum is
discrete
equation
e1e2  e’1  e’2
can not be satisfied exactly
Recall: in the Anderson model
the site-to-site hopping does
not conserve the energy
Decay rate of a quasiparticle with energy
e in 0d.
e
e
e’+ 
e’
Offdiagonal
matrix
element
M  , e , e  
1
g
 1
Chaos in Nuclei – Delocalization?
1
2
3
4
e
e’
e1’
....
5
6
generations
Can be mapped (approximately)
to the problem of localization
on Cayley tree
1
e1
Fermi Sea
Delocalization
in Fock space
2
3
4
5
Conventional Anderson Model
•one particle,
•one level per site,
•onsite disorder
•nearest neighbor hoping
Basis:
i , i
labels
sites
Hamiltonian: H
Hˆ 0   e i i i
i
 Hˆ 0  Vˆ
Vˆ 
I i
i , j  n. n.
j
e
0d system; no interactions
eb
many (N ) particles no interaction:
Individual energies e  and thus occupation
 
numbers n are conserved
e
conservation laws
“integrable system”
N
e
integrable system
Ĥ   E  

  n
 

E   n e 

0d system with interactions
e
Basis:
eb

n 
e
e

occupation
0,1 numbers
Hamiltonian:
Ĥ 0   E  

  n


labels
levels

H  Hˆ 0  Vˆ
Vˆ 
I

 
,
 
   
     .., n  1,.., n b  1,.., n  1,.., n  1,..
Conventional
Anderson
Model
Basis:
i
Hˆ   e i i i 
i
i , j  n .n .
I i
Basis:

i
labels
sites

Many body Andersonlike Model
j
labels
levels
 ,    n 

n  0,1
occupation
numbers
Ĥ   E   

I

 
,
 
   
“nearest
     .., n  1,.., n b  1,.., n  1,.., n  1,..
neighbors”:
Isolated quantum dot – 0d system of fermions
Exact many-body states:
Exact means that the imaginary
Ground state, excited states part of the energy is zero!
Quasiparticle excitations:
Finite decay rate
Q: What is the connection ?
S
gate
source
QD
drain
D
current
No e-e interactions –
resonance tunneling
S
gate
source
QD
drain
D
current
g
Mean level
spacing 1
No e-e interactions –
resonance tunneling
VSD
S
gate
source
QD
current
g
drain
D
The interaction leads to
No e-e interactions –
additional peaks –
resonance tunneling
many body excitations
VSD
S
S
D
Resonance tunneling
Peaks
D
Inelastic cotunneling
Additional peak
S
gate
source
QD
current
g
drain
S
The interaction leads to
additional peaks –
many body excitations
VSD
S
gate
source
QD
drain
D
current
g
Landau
quasiparticle with
the width SIA
NE
loc
Ergodic - WD
VSD
Landau
quasiparticle with
the width SIA
NE
loc
Localized - finite # of the satelites
Extended - infinite # of the satelites
Ergodic - WD
extended
VSD
(for finite e the number of the
satelites is always finite)
Ergodic – nonergodic crossover!
Anderson Model on a Cayley tree
Anderson Model on a Cayley tree
I, W
K – branching
number
W 1
Ic 
K ln K
W
I 
K
Resonance at
every generation
W
W
I
K ln K
K
Sparse
resonances
will call a quantum state 
Definition: We
ergodic if it occupies the number of
sites N  on the Anderson lattice,
which is proportional to the total
number of sites N :
N
N
 0

N
nonergodic
N
N
N
 const  0

ergodic
Localized states are
N




const

N


obviously not ergodic:
Q:
A:
Is each of the extended state ergodic
In 3D probably yes
For d>4 most likely no
?
nonergodic states
Such a state occupies infinitely
many sites of the Anderson model
but still negligible fraction of the
total number of sites
Example of nonergodicity: Anderson Model Cayley tree:
transition
K – branching number
W
Ic 
K ln K
ergodicity
n  ln N
I erg ~ W
crossover
I  W  K ln K 
Resonance is typically far
n  const
localized
W K  I  W  K ln K 
Typically there is no
resonance at the next step
n ~ ln N
nonergodic
n ~ ln N
nonergodic
W  I W K
Typically there is a
resonance at every step
I W
Typically each pair of nearest
neighbors is at resonance
n~ N
ergodic